arXiv subjects
Publications and source records attributed to Josefina Barrionuevo.
We construct families of $k$-step nilpotent symplectic Lie algebras associated with graphs, extending the construction given in [Pouseele-Tirao, JPAA 213 (2009)] for the 2-step case. We also show that, under mild conditions on the nilpotency type, there exist symplectic Lie algebras of that type.
We thoroughly explore the class of k-step nilpotent Lie algebras associated with a simple graph looking for k-step nilpotent Lie algebras which are rigid in the variety of at most k-step nilpotent Lie algebras. We find out that, besides the complete graph, the only examples arise for k=2 and graphs of at most 4 vertices. A key tool to prove non-k-rigidity in this context, is a general construction of non-trivial deformations for naturally graded nilpotent Lie algebras.
We present a novel construction of linear deformations for Lie algebras and use it to prove the non-rigidity of several classes of Lie algebras in different varieties. We consider the family of Lie algebras with an abelian factor showing that, in general, they are not rigid even for the case of a 1-dimensional abelian factor. We also address the problem of k-rigidity for k-step nilpotent Lie algebras. Using our construction and recent results in [BCC] we prove the that the k-step free nilpotent Lie algebras are k-rigid but not (k + 1)-rigid and that Heisenberg Lie algebras are 2-rigid but not 3-rigid.
We characterize those graphs which correspond to a rigid 2-step nilpotent Lie algebra in the variety of at most 2-step nilpotent Lie algebras.